Reedy fibrancy and composition in Segal spaces

Question

I am going through V. Hinich's "Lectures on Infinity Categories" and I have a (possibly trivial) question on Segal spaces.

We define Segal space to be a bisimplicial set $X$ which is fibrant in Reedy model structure and such that the "spine map" $X_n\to X_1\times_{X_0}X_1\times_{X_0}\ldots \times_{X_0}X_1$ is a weak equivalence of simplicial sets. Here by $X_i$ we mean $X_{i\ast}$.

Further on, when defining composition law in this model of infinity categories, we use the fact that $$ X_2\to X_1\times_{X_0}X_1 $$ is a trivial fibration.

The question is: that the last map is a weak equivalence is just the second axiom for Segal spaces. But how can I see that it is a fibration? I assume that it comes from the Reedy fibrancy assumption, but I fail to see how. Any help would be appreciated! If anybody knows a good reference, this will do the job too.

Answer 1

Given a simplicial object $X$ in a locally small category $\mathcal{M}$ and a simplicial set $S$, define the weighted limit $\{ S, X \}$ to be an object in $\mathcal{M}$ equipped with an isomorphism $$\textrm{Hom}_\mathcal{M} (-, \{ S, X \}) \cong \textrm{Hom}_{\textbf{sSet}} (S, \textrm{Hom}_\mathcal{M} (-, X))$$ of functors $\mathcal{M}^\textrm{op} \to \textbf{Set}$.

Lemma. Let $X$ be a simplicial object in a complete model category $\mathcal{M}$. The following are equivalent:

• $X$ is Reedy-fibrant.
• For every natural number $n$, the morphism $\{ \Delta^n, X \} \to \{ \partial \Delta^n, X \}$ induced by the inclusion $\partial \Delta^n \hookrightarrow \Delta^n$ is a fibration in $\mathcal{M}$.
• For every monomorphism $S \to T$ in $\textbf{sSet}$, the induced morphism $\{ T, X \} \to \{ S, X \}$ is a fibration in $\mathcal{M}$.

Proof. When $\mathcal{M}$ has limits for small diagrams, $\{ S, X \}$ exists for all simplicial sets $S$, so for fixed $X$, $\{ {-}, X \} : \textbf{sSet}^\textrm{op} \to \mathcal{M}$ is right adjoint to $\textrm{Hom}_\mathcal{M} (-, X) : \mathcal{M} \to \textbf{sSet}^\textrm{op}$. In particular, $\{ {-}, X \} : \textbf{sSet}^\textrm{op} \to \mathcal{M}$ takes colimits in $\textbf{sSet}$ to limits in $\mathcal{M}$. The second condition is now easily seen to be a paraphrase of the first condition, and the third condition is seen to be implied by the second condition by considering the skeletal filtration of a monomorphism.　◼

Thus, for example, if $X$ is a Reedy-fibrant bisimplicial set then the morphism you ask about is a Kan fibration because it is identifiable with morphism $\{ \Delta^2, X \} \to \{ \Lambda^2_1, X \}$ induced by the inner horn inclusion $\Lambda^2_1 \hookrightarrow \Delta^2$. Similarly for the spine maps: they are the morphisms induced by the inclusion of the spine into $\Delta^n$.

Answer 2

This is indeed a consequence of the Reedy fibrancy assumption: that says that $X_n \to X^{\partial \Delta^n}$ is a fibration for all $n$ (this is more or less [GJ99, VII.1.18 and Def. VII.2.1(2)]). But fibrations are stable under pullback [GJ99, Cor. II.1.3], so the pullback diagram $$\begin{array}{ccccc}X_2 & \to & X^{\partial \Delta^2} & \to & X^{\Lambda_1^2} \\ & & \downarrow & & \downarrow \\ & & X^{\Delta^{\{0,2\}}} & \to & X^{\partial \Delta^{\{0,2\}}}\end{array}$$ shows that $X_2 \to X^{\Lambda_1^2}$ is a fibration, since both $X_2 \to X^{\partial \Delta^2}$ and $X_1 \to X^{\partial \Delta^1}$ are.

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References.

[GJ99] P. G. Goerss, J. F. Jardine, Simplicial homotopy theory. Progress in Mathematics 174. Birkhäuser, 1999. ZBL0949.55001.